Discrete Elastic Rods Research Articles

Numerical exploration on buckling instability for directional control in flagellar propulsion.

We report a numerical method to control the swimming direction by exploiting buckling instability in uniflagellar bacteria and bio-inspired soft robots. Our model system is comprised of a spherical rigid head and a helical elastic flagellum. The rotation of the flagellum in low Reynolds environment generates a propulsive force that allows the system to swim in fluid. The locomotion is an intricate interplay between the elasticity of the flagellum, the hydrodynamic loading, and the flow generated by the moving head. We use the Discrete Elastic Rods algorithm to capture the geometrically nonlinear deformation in the flagellum, Lighthills Slender Body Theory to simulate the hydrodynamics, and Higdons model for the spherical head in motion within viscous fluid. This flagellated system follows a straight path if the angular velocity of the flagellum is below a critical threshold. Buckling ensues in the flagellum beyond this threshold angular velocity and the system takes a nonlinear trajectory. We consider the angular velocity as the control parameter and solve the inverse problem of computing the angular velocity, that varies with time, given a desired nonlinear trajectory. Our results indicate that bacteria can exploit buckling in flagellum to precisely control their swimming direction.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

TRAJECTORY SIMULATION OF SELF-PROPELLED ELASTIC RODS IN FLUID

The research for flexible structures coupling with fluid simulates and promotes the development of soft robotics. A fast and accurate numerical method is significant for a real-time simulation of robots. This research provides theoretical and critical information for experiments to reduce the possibility of failure, by anticipating the path of the underwater soft robots and the possible requiring parameters for materials. This paper researches for bio-swimming elastic rods coupling with two-dimensional incompressible Newton fluid. First, we discretize elastic rods to extensive springs and rotational springs and stablish the kinetic equations based on energy reflecting the influence of internal force on swimming rods, and solve the governing equation by leap-frog algorithm. Second, semi-Lagrangian method is used to establish a fluid solver. Finally, the simplified coupling algorithm based on immersed-boundary method is raised, calling immersed-boundary method for momentum equations. These equations update the velocity of grid near the coupling interface directly to replace the function source force play in immersed-boundary method. Combining the fluid solver, rods solver and the coupling algorithm, a integral program solving the problem of the dynamics for immersed rods numerically. Simulating the rods swimming with a referenced pose of sine curvature. Comparing the simulating result to existed experiments of filament swimming and to the swimming trajectory of soft-body fish, we find the numerical result conform to experiment result, leading to the conclusion that this algorithm and dynamic model can simulate the trajectory of discrete elastic rods swimming underwater smoothly under the influence of both internal elastic force and coupling soft body-fluid interaction. Using this program, we test several critical parameters relating to swimming performance of rods, including iteration numbers, frequency and initial phases, finding that changing the initial phase of rods will alter onward direction of elastic rods. These results prove the feasibility and possibility of this algorithm and program being used for guiding development of real soft swimmers.

Patterns of Carbon Nanotubes by Flow-Directed Deposition on Substrates with Architectured Topographies.

We develop and perform continuum mechanics simulations of carbon nanotube (CNT) deployment directed by a combination of surface topography and rarefied gas flow. We employ the discrete elastic rods method to model the deposition of CNT as a slender elastic rod that evolves in time under two external forces, namely, van der Waals (vdW) and aerodynamic drag. Our results confirm that this self-assembly process is analogous to a previously studied macroscopic system, the "elastic sewing machine", where an elastic rod deployed onto a moving substrate forms nonlinear patterns. In the case of CNTs, the complex patterns observed on the substrate, such as coils and serpentines, result from an intricate interplay between van der Waals attraction, rarefied aerodynamics, and elastic bending. We systematically sweep through the multidimensional parameter space to quantify the pattern morphology as a function of the relevant material, flow, and geometric parameters. Our findings are in good agreement with available experimental data. Scaling analysis involving the relevant forces helps rationalize our observations.

A Geometric Model for the Coiling of an Elastic Rod Deployed Onto a Moving Substrate

We report results from a systematic numerical investigation of the nonlinear patterns that emerge when a slender elastic rod is deployed onto a moving substrate; a system also known as the elastic sewing machine (ESM). The discrete elastic rods (DER) method is employed to quantitatively characterize the coiling patterns, and a comprehensive classification scheme is introduced based on their Fourier spectrum. Our analysis yields physical insight on both the length scales excited by the ESM, as well as the morphology of the patterns. The coiling process is then rationalized using a reduced geometric model (GM) for the evolution of the position and orientation of the contact point between the rod and the belt, as well as the curvature of the rod near contact. This geometric description reproduces almost all of the coiling patterns of the ESM and allows us to establish a unifying bridge between our elastic problem and the analogous patterns obtained when depositing a viscous thread onto a moving surface; a well-known system known as the fluid-mechanical sewing machine (FMSM).

Pattern morphology in the elastic sewing machine

We present results from a numerical investigation of the coiling patterns obtained when a slender elastic rod is deployed onto a moving substrate. The Discrete Elastic Rods method is employed to explore the parameter space, construct phase diagrams, identify their phase boundaries and characterize the pattern morphology. The various length scales of the patterns are primarily set by the gravity-bending length and depend only logarithmically on the deployment height. The curvature near the contact point, together with the dimensionless speed mismatch between deployment and the belt, dictate the characteristics of the patterns. The phase boundaries are found to be independent of both the gravito-bending length and the deployment height, as long as the latter is above a threshold value. We also evaluate the relative importance of twist and curvature strains, which confirms that bending has a dominant role.

Analysis of Bundle Formation in Biofilaments

Bundles of biofilaments are ubiquitous in cells with functions ranging from force transmission to cellular protection and thus understanding the self-assembly of the biofilaments into bundles is crucial. The conformation of a bundle is dictated by a small number of mechanical and chemical parameters. In this work, given a set of parameters, our aim is to find the most favorable bundle conformation and the number of filaments within. The filaments are treated as discrete elastic rods. The final bundle structure is reached as a result of the competition between the elastic energy and favorable chemical interaction among individual filaments in the bundle. The results indicate that there is a variety of different size bundles with different conformations attainable, from dimers to hexagonal-closed-packed. We discuss our results along with case studies of important biofilaments.

Discrete elastic rods

We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.